Concentration of least-energy solutions to a semilinear Neumann problem in thin domains

We consider the following semilinear elliptic equation: { − ε 2 Δ u + u − u p = 0 , u > 0 in Ω ε , ∂ u ∂ ν = 0 on ∂ Ω ε . Here, ε > 0 and p > 1 . Ω ε is a domain in R 2 with smooth boundary ∂ Ω ε , and ν denotes the outer unit normal to ∂ Ω ε . The domain Ω ε depends on ε, which shrinks to a straight line in the plane as ε → 0 . In this case, a least-energy solution exists for each ε sufficiently small, and it concentrates on a line. Moreover, the concentration line converges to the narrowest place of the domain as ε → 0 .